Problem: $ C = \left[\begin{array}{rrr}4 & -2 & 5 \\ -2 & 3 & 1\end{array}\right]$ $ A = \left[\begin{array}{rr}0 & 4 \\ 3 & -2 \\ 1 & 4\end{array}\right]$ What is $ C A$ ?
Explanation: Because $ C$ has dimensions $(2\times3)$ and $ A$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ C A = \left[\begin{array}{rrr}{4} & {-2} & {5} \\ {-2} & {3} & {1}\end{array}\right] \left[\begin{array}{rr}{0} & \color{#DF0030}{4} \\ {3} & \color{#DF0030}{-2} \\ {1} & \color{#DF0030}{4}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ A$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ A$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ A$ , and so on. Add the products together. $ \left[\begin{array}{rr}{4}\cdot{0}+{-2}\cdot{3}+{5}\cdot{1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{0}+{-2}\cdot{3}+{5}\cdot{1} & ? \\ {-2}\cdot{0}+{3}\cdot{3}+{1}\cdot{1} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{4}\cdot{0}+{-2}\cdot{3}+{5}\cdot{1} & {4}\cdot\color{#DF0030}{4}+{-2}\cdot\color{#DF0030}{-2}+{5}\cdot\color{#DF0030}{4} \\ {-2}\cdot{0}+{3}\cdot{3}+{1}\cdot{1} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{4}\cdot{0}+{-2}\cdot{3}+{5}\cdot{1} & {4}\cdot\color{#DF0030}{4}+{-2}\cdot\color{#DF0030}{-2}+{5}\cdot\color{#DF0030}{4} \\ {-2}\cdot{0}+{3}\cdot{3}+{1}\cdot{1} & {-2}\cdot\color{#DF0030}{4}+{3}\cdot\color{#DF0030}{-2}+{1}\cdot\color{#DF0030}{4}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-1 & 40 \\ 10 & -10\end{array}\right] $